# Predicting long-memory processes

I'm working with a two-state process with $x_t$ in $\{1, -1\}$ for $t = 1, 2, \ldots$

The autocorrelation function is indicative of a process with long-memory, i.e. it displays a power law decay with an exponent < 1. You can simulate a similar series in R with:

> library(fArma)
> x<-fgnSim(10000,H=0.8)
> x<-sign(x)
> acf(x)


My question: is there a canonical way to optimally predict the next value in the series given just the autocorrelation function? One way to predict is simply to use

$\hat{x}(t) = x(t-1)$

which has a classification rate of $(1 + \rho_1) / 2$, where $\rho$ is the lag-1 autocorrelation, but I feel like it must be possible to do better by taking into account the long-memory structure.

• I think part of the problem lies in the fact that the process you've laid out is not fully defined by the characteristics you've listed. For a sample of size $n$, you've given $n\choose 2$ linear constraints for $2^n$ parameters. Many processes could satisfy the constraints and yet lead to different achievable classification rates. Your $R$ code does uniquely define a process, but it seemed you intended that as a concrete example instead of as the main object of interest. Feb 12, 2011 at 4:43
• @cardinal, the problem should have to have known solution, which is probably found in W.Palma Long Memory time series: Theory and Methods. The point is that autocorrelation function may be used to obtain by Yule Walker system of equations the parameters of $AR(\infty)$ representation of the process, the point is when such representation exists (invertability) and what truncation is acceptable by the means of say MSE. For $R$ code in my PhD I used fracdiff package. Feb 14, 2011 at 12:21
• @Dmitrij, @Chris, the OP specifically states he is interested in binary-valued processes (I've got a pretty good guess at what he's likely interested in), for which an AR formulation via Yule-Walker would strike me as ad-hoc at the least. Perhaps you could throw a logistic around it to estimate a conditional probability, but it's still important to recognize the assumptions one is making in that case. Also, for long-memory processes, the choice of truncation can be important and induce nontrivial artifacts. Feb 14, 2011 at 16:16
• @cardinal, @Chris. oh, I as usually missed the part of the task ^__^ In the case of binary-valued process it seems to be a very well known (studied) problem of traffic measurement that comes from communication networks or so called ON/OFF process that exhibits long range dependence (long memory) property. As for the particular example, I'm a bit confused, since in "one way to predict" Chris actually takes the previous value, not using the ACF only (or I'm even more confused by the term "classification rate"). Feb 15, 2011 at 8:44
• I imagine it would be possible to take the code for an autoregressive fractionally integrated model and change the likelihood function to incorporate a probit effects. Then you could get the probability of $1$ or $-1$.
– John
Aug 16, 2012 at 20:53